2008 Physics 2111 Fundamentals of Physics Chapter 10 1 Fundamentals of Physics Chapter 10 Rotation 1.Translation & Rotation 2.Rotational Variables Angular Position Angular Displacement Angular Velocity Angular Acceleration 3.Angular Vectors??? 4.Rotation & Constant Angular Acceleration 5.Relating Linear & Angular Variables 6.Kinetic Energy of Rotation 7.Calculating the Rotational Inertia 8.Torque 9.Newton’s 2 nd Law for Rotation 10.Work & Rotational Kinetic Energy Review & Summary Questions Exercises & Problems

2008 Physics 2111 Fundamentals of Physics Chapter 10 2 Motions Translation - motion along a straight line Rotation - motion about an axis Rigid Body - rotates without any change in shape Rotates with all its parts locked together Axis of Rotation (Rotation Axis) –Every point of the body moves in a circle centered on the axis

2008 Physics 2111 Fundamentals of Physics Chapter 10 3 Rotational Variables Reference Line - fixed in the body, rotating with the body, perpendicular to the rotation axis. Top View: 1 revolution = 360 O = 2  radians 1 radian = 57.3 O = revolution Units:  is in radians s and r in length units

2008 Physics 2111 Fundamentals of Physics Chapter 10 4 Rotational Variables All points in the disk move through the same angular displacement, d , but the distance that a typical point P i moves, ds i, depends on its distance from the center of rotation. Units:  is in radians 1 revolution = 360 O = 2  radians 1 radian = 57.3 O = revolution

2008 Physics 2111 Fundamentals of Physics Chapter 10 5 Rotational Variables Angular Acceleration Angular Displacement Angular Velocity x v a a la Chapter 2

2008 Physics 2111 Fundamentals of Physics Chapter gaps Shoot a long arrow through a narrow rotating gap 20 cm Independent of distance from center! r =.30 m, 8 gaps  = 2.5 rev/s 20 cm arrow minimum speed ?

2008 Physics 2111 Fundamentals of Physics Chapter 10 7 Angular Quantities as Vectors? Right Hand Rule - curl your right hand about the axis of rotation with your fingers pointing in the direction of rotation; your extended thumb points in the direction of the angular velocity vector. The angular velocity vector defines the axis of rotation, not a direction in which something moves.

2008 Physics 2111 Fundamentals of Physics Chapter 10 8 Right-Hand Rule Right Hand Rule - curl your right hand about the axis of rotation with your fingers pointing in the direction of rotation; your extended thumb points in the direction of the angular velocity vector. A Right Hand Screw advances in the direction of its angular velocity. The length of the angular velocity vector indicates the rate of rotation (radians per second) Angular Acceleration is similarly defined.

2008 Physics 2111 Fundamentals of Physics Chapter 10 9 Angular Vectors? Vectors can be added in any order; but this is not true for angular displacements. (more on this in the next chapter) “Rotation operations do not commute.” Angular displacements cannot be treated as vectors.

2008 Physics 2111 Fundamentals of Physics Chapter TABLE 11-1 Equations of Motion for Constant Linear Acceleration and for Constant Angular Acceleration Linear Equation Missing Variable Angular Equation v = v 0 + at x - x 0  -  0  =  0 +  t x - x 0 = v 0 t + at 2 v   -  0 =  0 t +  t 2 v 2 = v a(x - x 0 ) t t  2 =   (  -  0 ) x - x 0 = (v 0 + v)t a   -  0 = (  0 +  )t x - x 0 = vt - at 2 v 0  0  -  0 =  t -  t 2 Rotation with Constant Angular Acceleration

2008 Physics 2111 Fundamentals of Physics Chapter A wheel rotating with constant acceleration Starting from rest:  0 = 0 Constant acceleration:  = 2.00 rad/s 2 90 radians in a particular 3 second time interval How long turning before start of interval? Angular velocity at start ~ 14 revolutions

2008 Physics 2111 Fundamentals of Physics Chapter Units: (m/s) (rad/s) (m) Tangential: Velocity:Acceleration: Relating Angular & Linear Variables Radial:

2008 Physics 2111 Fundamentals of Physics Chapter Angular Acceleration & Angular Velocity are vectors: If the angular velocity is increasing (spinning faster & faster), the angular acceleration is in the same direction as the angular velocity. If the angular velocity is decreasing, the angular acceleration is... Tangential: Centripetal: Uniform Circular Motion:

2008 Physics 2111 Fundamentals of Physics Chapter Measure the speed of light 500 slots r = 5.0 cm L = 500 m c = 3 x 10 5 km/s , v t

2008 Physics 2111 Fundamentals of Physics Chapter wheels connected by a belt Every point of the belt is moving at the same velocity r a = 10 cm r c = 25 cm  a = 1.6 rad/s 2 How long for c to reach 100 rev/min?

2008 Physics 2111 Fundamentals of Physics Chapter Kinetic Energy of Rotation Consider a rotating body as a set of particles with different speeds. Then the kinetic energy of the body is: Or: (kg m 2 ) Rotational K.E. “Moment of Inertia” aka “Rotational Inertia”

2008 Physics 2111 Fundamentals of Physics Chapter Moment of Inertia Uniform rod of length L and mass M about a perpendicular axis through one end:

2008 Physics 2111 Fundamentals of Physics Chapter Rotational Inertia: Uniform hoop of radius R and mass M about a perpendicular axis through its center: Uniform disk of radius R and mass M about a perpendicular axis through its center:

2008 Physics 2111 Fundamentals of Physics Chapter Some Moments of Inertia

2008 Physics 2111 Fundamentals of Physics Chapter Calculating Moments of Inertia Parallel Axis Theorem:

2008 Physics 2111 Fundamentals of Physics Chapter Parallel Axis Theorem: Calculating Moments of Inertia

2008 Physics 2111 Fundamentals of Physics Chapter Rank rotational inertia about these axes: I max I min Parallel Axis Theorem :

2008 Physics 2111 Fundamentals of Physics Chapter Rotational Inertia = ? Table 11-2: Parallel Axis Theorem:

2008 Physics 2111 Fundamentals of Physics Chapter Torque - a force causes an object to rotate

2008 Physics 2111 Fundamentals of Physics Chapter Vector (Cross) Product Magnitude: “Anti-commutative”: c x = a y b z - b y a z c y = a z b x - b z a x c z = a x b y - b x a y Direction: “the right-hand rule”

2008 Physics 2111 Fundamentals of Physics Chapter The Vector Cross Product

2008 Physics 2111 Fundamentals of Physics Chapter Newton’s 2 nd Law for Rotation Consider a point rotating about an axis:

2008 Physics 2111 Fundamentals of Physics Chapter rod is not to turn

2008 Physics 2111 Fundamentals of Physics Chapter Work & Rotational Kinetic Energy Consider a point rotating about an axis: Work – Kinetic Energy Theorem: Power in rotating an object about a fixed axis:

2008 Physics 2111 Fundamentals of Physics Chapter TABLE 11-3 Some Corresponding Relations for Translational and Rotational Motion Pure Translation (Fixed Direction) Pure Rotation (Fixed Axis) Position x Angular position  Velocity v = dx/dy Angular velocity  = d  /dt Acceleration a = dv/dt Angular acceleration  = d  /dt Mass m Rotational inertia I Newton's second law F net = ma Newton's second law  net = I  Work F dx Work  d  Kinetic energy mv 2 Kinetic energy I  2 Power (constant force) P = Fv Power (constant torque) P =  Work–kinetic energy theorem W =  K Work–kinetic energy theorem W =  K Corresponding Relationships

2008 Physics 2111 Fundamentals of Physics Chapter Massless cord wrapped around a massive wheel T R = 0.2 m I = 0.05 kg m 2 M = 2 kg P = 3 N  a  = ? M = 2 kg P = 3 N TT

2008 Physics 2111 Fundamentals of Physics Chapter The box descends after being released R = 0.20 m I = 0.4 kg m 2 KE B = 6 J KE W = ? h = ? K box = 6 JBox